A Time-Domain Model Order Reduction Method for Power Systems Based on Laguerre Polynomials and Gram Matrices

Abstract

The present application provides a time-domain model reduction method for power systems based on Laguerre polynomials and an approximate Gram matrix. It can efficiently reduce the state-space model of a power system by combining Laguerre polynomials and an approximate Gram matrix, and use the reduced-order model to accurately predict the system response of the power system. On the one hand, during the reduction process, the present application no longer needs to solve two large Lyapunov equations to calculate the controllability Gram matrix and the observability Gram matrix. Instead, it uses Laguerre orthogonal polynomials to perform low-rank approximation on the controllability Gram matrix and the observability Gram matrix of the power system, thereby reducing the computational complexity and achieving efficient model reduction, and further improving the power system response prediction efficiency. On the other hand, the reduced-order model of the present application has a global error bound, thus improving the accuracy of power system response prediction. It can be seen that the present application can balance good prediction accuracy and low computational complexity.

Description

Technical Field

[0001] The present application relates to the technical field of power systems, and particularly to a method for reducing the order of a time-domain model of a power system based on Laguerre polynomials and Gram matrices. Background Art

[0002] As the power system continues to develop towards large-scale configuration, the system scale of the power system can reach thousands of buses and dynamic components. The model equations used to describe the time-domain response of the power system involve nonlinear ordinary differential equations, and the dimensions of these equations after linearization at the equilibrium point appear in large numbers. To better analyze the small-signal stability of the power system, it is necessary to find a method that can reduce the model scale, data storage requirements, and computational complexity of the time-domain model of the power system. Thus, reducing the order of the time-domain model of the power system is an important part of linear control system analysis and power system dynamic model design, and can be used to effectively reduce the computational amount of controller design.

[0003] Currently, the methods for reducing the order of the time-domain model of the power system mainly include moment matching method, modal truncation method, and balanced truncation method. Among them, in the moment matching method, the system transfer function of the time-domain model of the power system can be expanded around an operating point, and a Krylov subspace technique can be used to construct a reduced-order system. Although this is an effective method, it does not have a global error bound. The modal truncation method retains the dominant modes of the system and truncates the modes with lower importance, and cannot completely retain the input and output characteristics of the power system. The balanced truncation method is well-known for its ability to capture the input-output behavior and dominant modes of the power system. Compared with other order reduction methods, the balanced truncation method has theoretical advantages, including global error bounds and stability preservation. However, when using the exact balanced truncation method to reduce the order of the time-domain model of the power system, two large Lyapunov equations need to be solved to obtain the controllable Gram matrix and the observable Gram matrix, which results in significant computational complexity.

[0004] Thus, the prior art cannot balance better response prediction accuracy and lower computational complexity. Summary of the Invention

[0005] The purpose of the present application aims to at least solve one of the above technical defects, especially the technical defect that the prior art cannot balance better response prediction accuracy and lower computational complexity.

[0006] In one embodiment, the present application provides a method for reducing the order of a time-domain model of a power system based on Laguerre polynomials and Gram matrices, including:

[0007] Obtain the state-space model of the power system;

[0008] Using the Laguerre polynomial approximation algorithm, calculate each first Laguerre coefficient corresponding to the state impulse response of the state space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model; wherein, the dual model is dual to the state space model.

[0009] Calculate the first low-rank factorization factor matrix corresponding to the controllability Gram matrix according to each of the first Laguerre coefficients, and calculate the second low-rank factorization factor matrix corresponding to the observability Gram matrix according to each of the second Laguerre coefficients.

[0010] Based on the first low-rank factorization factor matrix and the second low-rank factorization factor matrix, obtain the reduced-order model corresponding to the state space model.

[0011] In response to a system response prediction request, receive the input vector to be predicted of the power system, and predict the time-domain response vector of the power system according to the input vector to be predicted and the reduced-order model.

[0012] In one embodiment, the state space model includes a system matrix, an input matrix, and an output matrix.

[0013] The step of using the Laguerre polynomial approximation algorithm to calculate each first Laguerre coefficient corresponding to the state impulse response of the state space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model includes:

[0014] Based on the following expression, calculate each of the first Laguerre coefficients:

[0015]

[0016] In the formula, f0 is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the state space model; f i is the coefficient of the (i + 1)-th Laguerre expansion corresponding to the state impulse response of the state space model; is the n1-dimensional identity matrix, n1 is the number of differential variables in the linear differential-algebraic equation of the power system, and the state space model is the equivalent model of the linear differential-algebraic equation; A is the system matrix, B is the input matrix, and l is the pre-determined number of Laguerre expansion terms.

[0017] Based on the following expression, calculate each of the second Laguerre coefficients:

[0018]

[0019] where \(g_0\) is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the dual model; \(g\) i is the coefficient of the \((i + 1)\)-th Laguerre expansion corresponding to the state impulse response of the dual model; \(A\) T is the transpose matrix of \(A\), \(C\) T is the transpose matrix of \(C\), and \(C\) is the output matrix.

[0020] In one embodiment, calculating the first low-rank decomposition factor matrix corresponding to the controllability Gram matrix according to each of the first Laguerre coefficients, and calculating the second low-rank decomposition factor matrix corresponding to the observability Gram matrix according to each of the second Laguerre coefficients includes:

[0021] Calculating the integral result of the product of Laguerre polynomials within a preset time interval, and constructing an integral matrix according to the integral result;

[0022] Performing Cholesky decomposition on the integral matrix to obtain a first lower triangular matrix;

[0023] Obtaining the first low-rank decomposition factor matrix according to the first lower triangular matrix and each of the first Laguerre coefficients;

[0024] Obtaining the second low-rank decomposition factor matrix according to the first lower triangular matrix and each of the second Laguerre coefficients.

[0025] In one embodiment, the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix are obtained based on the following expressions:

[0026]

[0027] where \(R\) is the first low-rank decomposition factor matrix; \(F\) is the coefficient matrix composed of each of the first Laguerre coefficients; \(J\) is the first lower triangular matrix; \(I\) p is the \(p\)-dimensional identity matrix, and \(p\) is the number of inputs of the power system; \(L\) is the second low-rank decomposition factor matrix; \(G\) is the coefficient matrix composed of each of the second Laguerre coefficients; \(I\) q is the \(q\)-dimensional identity matrix, and \(q\) is the number of outputs of the power system; is the Kronecker product operator.

[0028] In one embodiment, obtaining the reduced-order model corresponding to the state-space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix includes:

[0029] Performing an operation on \(L\)T The R matrix is subjected to SVD decomposition, and according to the SVD decomposition result and a preset order reduction order, a left truncated matrix U r , a diagonal truncated matrix Σ r and a right truncated matrix V r are obtained respectively; where R is the first low-rank decomposition factor matrix, and L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix;

[0030] According to the left truncated matrix U r , the diagonal truncated matrix Σ r and the right truncated matrix V r , the left projection matrix and the right projection matrix are calculated respectively;

[0031] According to the left projection matrix and the right projection matrix, the order reduction model is obtained.

[0032] In one embodiment, the left projection matrix and the right projection matrix are obtained based on the following expressions:

[0033]

[0034] In the formula, W l is the left projection matrix; Σ r is the diagonal truncated matrix; is the transpose matrix of U r , U r is the left truncated matrix; L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix; W r is the right projection matrix; R is the first low-rank decomposition factor matrix; V r is the right truncated matrix.

[0035] In one embodiment, the state space model includes a system matrix A, an input matrix B, an output matrix C, and a constant matrix D;

[0036] The obtaining of the order reduction model according to the left projection matrix and the right projection matrix includes:

[0037] Based on the following expressions, a reduced-order system matrix A r , a reduced-order input matrix B r and a reduced-order output matrix C r are obtained respectively:

[0038] A r =W l AW r , B r =W l B, C r= CW r , D r = D

[0039] wherein, W l is the left projection matrix, and W r is the right projection matrix;

[0040] Construct the reduced-order model according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix, and the constant matrix.

[0041] In a second aspect, an embodiment of the present application provides a power system time-domain model reduction device based on Laguerre polynomials and Gram matrices, including:

[0042] A state space model acquisition module, configured to acquire the state space model of the power system;

[0043] A Laguerre coefficient acquisition module, configured to use the Laguerre polynomial approximation algorithm to calculate each first Laguerre coefficient corresponding to the state impulse response of the state space model, and each second Laguerre coefficient corresponding to the state impulse response of the dual model; wherein, the dual model is dual to the state space model;

[0044] A low-rank decomposition module, configured to calculate a first low-rank decomposition factor matrix corresponding to the controllable Gram matrix according to each of the first Laguerre coefficients, and calculate a second low-rank decomposition factor matrix corresponding to the observable Gram matrix according to each of the second Laguerre coefficients;

[0045] A reduction module, configured to obtain a reduced-order model corresponding to the state space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix;

[0046] A response prediction module, configured to, in response to a system response prediction request, receive the input vector to be predicted of the power system, and predict the time-domain response vector of the power system according to the input vector to be predicted and the reduced-order model.

[0047] In a third aspect, an embodiment of the present application provides a storage medium, in which computer-readable instructions are stored, and when the computer-readable instructions are executed by one or more processors, the one or more processors are caused to execute the steps of the power system time-domain model reduction method based on Laguerre polynomials and Gram matrices in any of the above embodiments.

[0048] In a fourth aspect, an embodiment of the present application provides a computer device, which includes: one or more processors, and a memory;

[0049] The memory stores computer-readable instructions that, when executed by the one or more processors, perform the steps of the method for reducing the order of the time-domain model of the power system based on Laguerre polynomials and the Gram matrix described in any of the above embodiments.

[0050] In the method, apparatus, storage medium, and computer device for reducing the order of the time-domain model of the power system based on Laguerre polynomials and the Gram matrix provided in some embodiments of the present application, the state-space model of the power system can be efficiently reduced by combining Laguerre polynomials and the Gram matrix, and the system response of the power system can be accurately predicted using the reduced-order model. On the one hand, in the process of reducing the order, it is no longer necessary to solve two large Lyapunov equations to calculate the controllability Gram matrix and the observability Gram matrix. Instead, the Laguerre orthogonal polynomials are used to perform low-rank approximation on the controllability Gram matrix and the observability Gram matrix of the power system, thereby reducing the computational complexity and achieving efficient order reduction, and further improving the efficiency of power system response prediction. On the other hand, the reduced-order model of the present application has a global error bound, thereby improving the accuracy of power system response prediction. It can be seen that the present application can balance good prediction accuracy and low computational complexity. Description of the Drawings

[0051] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0052] Figure 1 is a schematic flowchart of the method for reducing the order of the time-domain model of the power system based on Laguerre polynomials and the Gram matrix in one embodiment;

[0053] Figure 2 is a schematic flowchart of the Laguerre coefficient calculation step in one embodiment;

[0054] Figure 3 In one embodiment, when the input vector u(t) = e -0.1t sin(t), it is a schematic diagram of the output responses of the LRBT reduced-order model and the BT reduced-order model;

[0055] Figure 4 In one embodiment, when the input vector u(t) = e -0.1t sin(t), it is a schematic diagram of the response errors of the LRBT reduced-order model and the BT reduced-order model;

[0056] Figure 5 In an embodiment, when the input vector u(t) = sin(0.5t), the schematic diagram of the output responses of the LRBT reduced-order model and the BT reduced-order model;

[0057] Figure 6 In an embodiment, when the input vector u(t) = sin(0.5t), the schematic diagram of the response errors of the LRBT reduced-order model and the BT reduced-order model;

[0058] Figure 7 In an embodiment, the comparison chart of the prediction time-consuming of the LRBT reduced-order model and the BT reduced-order model;

[0059] Figure 8 In an embodiment, the structural schematic diagram of the power system time-domain model reduction device based on Laguerre polynomials and Gram matrix;

[0060] Figure 9 In an embodiment, the internal structure diagram of the computer device. Specific implementation manners

[0061] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present application.

[0062] In an embodiment, as Figure 1 shown, the present application provides a power system time-domain model reduction method based on Laguerre polynomials and Gram matrix, which specifically includes the following steps:

[0063] S102: Obtain the state-space model of the power system.

[0064] Among them, the state-space model is a dynamic time-domain model for predicting the change of the response vector of the power system over time. In the present application, the state-space model of the power system can be an ordinary differential system with an index of 1.

[0065] In this step, for the full-system linear differential-algebraic equation of the power system under the small-signal stability analysis problem, an equivalent state-space model can be constructed to describe the relationship between the input and the response through the state-space model.

[0066] Specifically, if the circuit system has p inputs and q outputs, the linearized model of the power system can be represented by the following n-order linear time-invariant differential-algebraic equation Σ:

[0067]

[0068] where ε is the input constant matrix, is the first-order differential of is the system matrix of the linear differential-algebraic equation, is the generalized state vector, is the input matrix of the linear differential-algebraic equation, u(t) is the input vector, and y(t) is the response vector, is the output matrix of the linear differential-algebraic equation, is the constant matrix of the linear differential-algebraic equation. is the set of real numbers.

[0069] Among them, and have the following block structure:

[0070]

[0071] where is the n1-dimensional identity matrix. n1 is the number of differential variables in the linear differential-algebraic equation of the power system, n2 is the number of algebraic variables in the linear differential-algebraic equation of the power system, and n1 + n2 = n. and are both submatrices of and are both submatrices of and are both submatrices of and are both submatrices of is the state vector of the power system, is the operating vector of the power system.

[0072] When is non-singular, the system is a differential-algebraic model with index 1, and it can be rewritten as an equivalent ordinary differential system in the following form:

[0073]

[0074] where is the first-order differential of

[0075] Since in the actual power system, It is usually non-singular, so the running vector in the above equation can be eliminated. Let the state variable Furthermore, an n1-order state space model equivalent to the system Σ is obtained, that is, the ordinary differential system Σ s :

[0076]

[0077] Where:

[0078]

[0079] In the formula, is the first-order differential of x(t), x(t) is the generalized state vector of the state space model, A is the system matrix of the state space model, B is the input matrix of the state space model, C is the output matrix of the state space model, D is the constant matrix of the state space model, u(t) is the input vector, and y(t) is the response vector.

[0080] S104: Adopt the Laguerre polynomial approximation algorithm to calculate each first Laguerre coefficient corresponding to the state impulse response of the state space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model respectively; wherein, the dual model is dual to the state space model.

[0081] In this step, the Laguerre polynomial can be used to approximately expand and solve the expansion coefficients of the state variables in the state space model ∑ s The expansion coefficient is the Laguerre coefficient described in the embodiments of the present application.

[0082] Specifically, the definition of the Laguerre polynomial is:

[0083]

[0084] In the formula, L0(t) is the first Laguerre expansion term, L i (t) is the (i + 1)-th Laguerre expansion term, e is the natural constant, t is the time, and d is the differential symbol.

[0085] Its three-term recurrence formula is:

[0086]

[0087] The Laguerre polynomial also has integral recurrence properties:

[0088]

[0089] In the formula, τ is the variable symbol.

[0090] The Laguerre polynomial sequence {L i (t)} forms a uniformly bounded orthogonal basis of the Hilbert space , so for any function it can be expanded into the following Laguerre polynomial series:

[0091]

[0092] where F i is the Laguerre coefficient.

[0093] In this step, let the input vector of the state space model ∑ s be the unit impulse function δ(t), then the corresponding state equation is:

[0094]

[0095] Integrate both sides of the above state equation over [0, t], that is:

[0096]

[0097] Assume the initial condition is zero and use the integral property of the unit impulse function At this time, the above equation can be changed to:

[0098]

[0099] Perform a Laguerre polynomial approximation expansion on the state variable of the above equation, that is, the state impulse response of the power system:

[0100]

[0101] where l is the number of Laguerre expansion terms, and its specific value can be obtained according to the actual situation. In an example, 10 ≤ l ≤ 30, and its specific value can be increased or decreased as appropriate according to the actual situation. f i is the i-th first Laguerre coefficient.

[0102] Similarly, the dual model of the state space model ∑ s is:

[0103]

[0104] where is the first derivative of z(t), z(t) is the generalized state vector of the dual model. A T is the transpose matrix of A, B T is the transpose matrix of B, C T is the transpose matrix of C. ud (t) is the input vector of the dual model, y d (t) is the response vector of the dual model.

[0105] Let the input vector of the dual model be the unit impulse function δ(t) to obtain the state variables of the dual model, that is, the state impulse response of the dual model. Perform Laguerre polynomial approximation expansion on the state impulse response of the dual model:

[0106]

[0107] In the formula, g i is the i-th second Laguerre coefficient.

[0108] S106: Calculate the first low-rank decomposition factor matrix corresponding to the controllable Gram matrix according to each first Laguerre coefficient, and calculate the second low-rank decomposition factor matrix corresponding to the observable Gram matrix according to each second Laguerre coefficient.

[0109] In this step, for the state space model Σ of the power system s :(A, B, C, D), give the controllable Gram matrix P and the observable Gram matrix Q in the time interval [0, T]:

[0110]

[0111] Based on the specific forms of the controllable Gram matrix P and the observable Gram matrix Q. Among them, e At B is the state impulse response corresponding to the state space model Σ s when the input vector is the unit impulse function. They are respectively the state impulse responses corresponding to the dual model when the input vector is the unit impulse function. Therefore, the controllable Gram matrix P and the observable Gram matrix Q can be rewritten by using the above x(t) and z(t):

[0112]

[0113] In the formula, x(t) T is the transpose matrix of x(t), and z(t) T is the transpose matrix of z(t).

[0114] Substitute the Laguerre polynomial approximations of x(t) and z(t) into the above formula to obtain:

[0115]

[0116] In the formula, is The transposed matrix of is the transposed matrix of

[0117] It can be seen that in this step, the controllable Gram matrix can be approximately expressed according to each first Laguerre coefficient, and then the first low-rank decomposition factor matrix corresponding to the controllable Gram matrix can be obtained. Similarly, in this step, the observable Gram matrix can be approximately expressed according to each second Laguerre coefficient, and then the second low-rank decomposition factor matrix corresponding to the observable Gram matrix can be obtained.

[0118] S108: Based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix, obtain the reduced-order model corresponding to the state-space model.

[0119] In this step, after obtaining the low-rank decomposition factor matrices corresponding to the controllable Gram matrix and the observable Gram matrix, the state-space model of the power system can be reduced in order based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix to obtain the reduced-order model.

[0120] S110: In response to a system response prediction request, receive the input vector to be predicted of the power system, and predict the time-domain response vector of the power system according to the input vector to be predicted and the reduced-order model.

[0121] In this step, after obtaining the reduced-order model of the power system, the reduced-order model can be used to predict the response of the power system. During the prediction process, the input vector to be predicted of the power system can be received and used as u(t) in the reduced-order model to predict the time-domain response vector of the power system through the reduced-order model, thereby realizing response prediction.

[0122] In this embodiment, the computational complexity of response prediction mainly comes from the solution of Laguerre coefficients. Since the solution of Laguerre coefficients is a simple iterative calculation, the computational complexity can be greatly reduced and the memory occupancy can be reduced. In this way, the efficiency of predicting the response can be improved. At the same time, the reduced-order model of this application has a global error bound, thereby improving the accuracy of power system response prediction. It can be seen that this application can balance good prediction accuracy and low computational complexity.

[0123] In one embodiment, the state-space model includes a system matrix A, an input matrix B, and an output matrix C. For the relevant descriptions of the system matrix A, the input matrix B, and the output matrix C, reference can be made to the above embodiments, which will not be elaborated herein.

[0124] In this embodiment, each first Laguerre coefficient can be calculated respectively based on the following expression:

[0125]

[0126] In the formula, f0 is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the state space model; f i is the coefficient of the (i + 1)-th Laguerre expansion corresponding to the state impulse response of the state space model; is an n1-dimensional identity matrix, n1 is the number of differential variables in the linear differential-algebraic equation of the power system, and the state space model is an equivalent model of the linear differential-algebraic equation; A is the system matrix, B is the input matrix, and l is the pre-determined number of Laguerre expansion terms.

[0127] In this embodiment, each second Laguerre coefficient can be calculated respectively based on the following expression:

[0128]

[0129] In the formula, g0 is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the dual model; g i is the coefficient of the (i + 1)-th Laguerre expansion corresponding to the state impulse response of the dual model; A T is the transpose matrix of A, C T is the transpose matrix of C, and C is the output matrix.

[0130] Specifically, after the state impulse response of the state space model is approximately expanded by the Laguerre polynomial as , it can be substituted into the integral equation to obtain:

[0131]

[0132] Using the integral recursive property of the Laguerre polynomial, the above formula can be changed to:

[0133]

[0134] Eliminating L i (t) (i = 0, 1,..., l - 1) on both sides of the equation, the first Laguerre coefficient f i can be obtained as:

[0135]

[0136] In the formula, is 's inverse matrix.

[0137] It can be determined through the specific forms of the state space model and the dual model that the second Laguerre coefficient is actually obtained by replacing A in the first Laguerre coefficient with A T , and replacing B with C T . Therefore, the second Laguerre coefficient g i is:

[0138]

[0139] Based on the expressions of the first Laguerre coefficient f i and the second Laguerre coefficient g i , it can be determined that the solution of the first Laguerre coefficient f i and the second Laguerre coefficient g i can be assisted by the state equation and the recurrence property of the Laguerre polynomial. In an example, the calculation steps of the Laguerre coefficient can be as Figure 2 shown.

[0140] In one embodiment, calculating the first low-rank decomposition factor matrix corresponding to the controllable Gram matrix according to each first Laguerre coefficient, and calculating the second low-rank decomposition factor matrix corresponding to the observable Gram matrix according to each second Laguerre coefficient, includes:

[0141] Step A1: Calculate the integral result of the product of the Laguerre polynomials within a preset time interval, and construct an integral matrix according to the integral result;

[0142] Step A3: Perform Cholesky decomposition on the integral matrix to obtain a first lower triangular matrix;

[0143] Step A5: Obtain the first low-rank decomposition factor matrix according to the first lower triangular matrix and each first Laguerre coefficient;

[0144] Step A7: Obtain the second low-rank decomposition factor matrix according to the first lower triangular matrix and each second Laguerre coefficient.

[0145] In this embodiment, in the process of performing low-rank approximate decomposition on the controllable Gram matrix and the observable Gram matrix, first, a matrix with the integral of the product of the Laguerre polynomials as elements on a given time interval and its Cholesky decomposition can be calculated.

[0146] Specifically, on a given time interval [0, T], the integral of the product of the Laguerre polynomials L i (t) and L j (t) is:

[0147]

[0148] where \(0\leq i\leq l - 1\), \(0\leq j\leq l - 1\). Define the integral matrix \(H\) i+1,j+1 with elements The subscript \(i + 1\) represents the row number of the element in the integral matrix, and the subscript \(j + 1\) represents the column number of the element in the integral matrix. If the integral matrix \(H\) is positive definite, the Cholesky decomposition can be directly performed on the integral matrix \(H\):

[0149] \(H = JJ\) T

[0150] where \(J\) is the first lower triangular matrix.

[0151] If the integral matrix \(H\) is not positive definite, a damping factor \(\mu=\|H\|_2\) can be introduced to correct the integral matrix. During the correction, let such that the resulting matrix is positive definite, and then perform the Cholesky decomposition on the resulting matrix to obtain the first lower triangular matrix. Among them, \(I\) l is the \(l\)-dimensional identity matrix.

[0152] After obtaining the first lower triangular matrix, the Laguerre approximation expansion result and the first lower triangular matrix can be used to calculate the first low-rank decomposition factor matrix corresponding to the controllable Gram matrix and the second low-rank decomposition factor matrix corresponding to the observable Gram matrix, respectively.

[0153] Furthermore, if it is represented in matrix form and then we can get:

[0154]

[0155] where \(I\) p is the \(p\)-dimensional identity matrix, \(p\) is the number of inputs of the power system; \(L\) is the second low-rank decomposition factor matrix. \(I\) q is the \(q\)-dimensional identity matrix, \(q\) is the number of outputs of the power system.

[0156] Let the matrix \(F = [f_0\ f_1\cdots f\) l-1 , \(G = [g_0\ g_1\cdots g\) l-1 . Combining with the specific form of the integral matrix \(H\), the above formula can be further rewritten as:

[0157]

[0158] where \(P\) is the controllable Gram matrix, \(F\) T is the transpose matrix of \(F\), \(Q\) is the observable Gram matrix, \(G\) T is the transpose matrix of \(G\). is the Kronecker product operator.

[0159] Since the integral matrix H has a Cholesky decomposition H = JJ T , and it is not difficult to obtain:

[0160]

[0161] where

[0162] then the low-rank decomposition form of the controllability Gram matrix P and the low-rank decomposition form of the observability Gram matrix Q can be:

[0163]

[0164] The first low-rank decomposition factor matrix R and the second low-rank decomposition factor matrix L satisfy:

[0165]

[0166] where R is the first low-rank decomposition factor matrix; F is the coefficient matrix composed of each first Laguerre coefficient; J is the first lower triangular matrix; I p is the p-dimensional identity matrix, and p is the number of inputs of the power system; L is the second low-rank decomposition factor matrix; G is the coefficient matrix composed of each second Laguerre coefficient; I q is the q-dimensional identity matrix, and q is the number of outputs of the power system; is the Kronecker product operator.

[0167] In one embodiment, based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix, a reduced-order model corresponding to the state-space model is obtained, including:

[0168] Step B1: Perform SVD decomposition on the L T R matrix, and respectively obtain the left truncation matrix U r , the diagonal truncation matrix Σ r and the right truncation matrix V r according to the SVD decomposition result and the preset reduced-order number; where R is the first low-rank decomposition factor matrix, and L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix;

[0169] Step B3: Calculate the left projection matrix and the right projection matrix respectively according to the left truncation matrix U r , the diagonal truncation matrix Σ r and the right truncation matrix V r ;

[0170] Step B5: Obtain a reduced-order model based on the left projection matrix and the right projection matrix.

[0171] In this embodiment, the state-space model can be reduced in order based on the approximate balanced truncation algorithm (abbreviated as the LRBT algorithm) to obtain a reduced-order model ∑. r In this way, when predicting the response of the power system using the reduced-order model, the prediction accuracy can be further improved.

[0172] Specifically, after obtaining the first low-rank factorization factor matrix R and the second low-rank factorization factor matrix L, the L T R matrix can be processed by SVD (Singular Value Decomposition) to obtain the left singular vector matrix U, the diagonal matrix Σ, and the right singular vector matrix V, where L T R = UΣV T Here, V T is the transpose matrix of V.

[0173] If the reduction order is r, the left singular vector matrix U, the diagonal matrix Σ, and the right singular vector matrix V can be intercepted respectively according to the reduction order to obtain the left interception matrix U r , the diagonal interception matrix Σ r , and the right interception matrix V r . In one example, the left interception matrix U r is the first r columns of the left singular vector matrix U, that is, U r = U(:,1:r). The right interception matrix V r is the first r columns of the right singular vector matrix V, that is, V r = V(:,1:r). The diagonal interception matrix Σ r is the leading r-order principal minor of the diagonal matrix Σ, that is, Σ r = Σ(1:r,1:r).

[0174] It can be understood that the reduction order r can be determined in advance according to actual factors such as the computational complexity requirement and the computational accuracy requirement, and no specific limitation is made herein.

[0175] After obtaining the left interception matrix U r , the diagonal interception matrix Σ r , and the right interception matrix V r , the left projection matrix W r and the right projection matrix W r can be calculated respectively according to the left interception matrix U r , the diagonal interception matrix Σ l , and the right interception matrix V r , and the state-space model can be reduced in order based on this, and then a reduced-order model is obtained.

[0176] In one embodiment, the left projection matrix and the right projection matrix are obtained based on the following expressions:

[0177]

[0178] where W l is the left projection matrix; Σ r is the diagonal truncation matrix; U r T is the transpose matrix of U r , U r is the left truncation matrix; L T is the transpose matrix of L, and L is the second low-rank factorization factor matrix; W r is the right projection matrix; R is the first low-rank factorization factor matrix; V r is the right truncation matrix.

[0179] In one embodiment, the state space model includes a system matrix A, an input matrix B, an output matrix C, and a constant matrix D. For the relevant descriptions of the system matrix A, the input matrix B, the output matrix C, and the constant matrix D, reference may be made to the descriptions of the above embodiments, which will not be elaborated herein.

[0180] Based on the left projection matrix and the right projection matrix, a reduced-order model is obtained, including:

[0181] Step B51: The reduced-order system matrix A r , the reduced-order input matrix B r , and the reduced-order output matrix C r are respectively obtained based on the following expressions:

[0182] A r = W l AW r , B r = W l B, C r = CW r , D r = D

[0183] where W l is the left projection matrix, and W r is the right projection matrix;

[0184] Step B53: A reduced-order model is constructed according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix, and the constant matrix.

[0185] In this embodiment, a reduced-order model Σ r : (A r , B r , C r , D r ) can be constructed according to the left projection matrix and the right projection matrix. Among them, the reduced-order model Σr It can be:

[0186]

[0187] In the formula, is the first-order differential of x r (t), x r (t) is the generalized state vector of the reduced-order model, u(t) is the input vector, and y r (t) is the response vector of the power system output by the reduced-order model.

[0188] The following are the performance verification results of the power system time-domain model reduction method based on Laguerre polynomials and Gram matrices provided by the embodiments of the present application.

[0189] An LRBT reduced-order model is generated by the method provided by the present application. Among them, the method provided by the present application can equivalently transform the original 13,251-dimensional differential-algebraic model into a state-space model containing 1,664 state variables and perform reduction to obtain the LRBT reduced-order model. In addition, an existing balanced truncation method is used to generate a BT reduced-order model. The LRBT reduced-order model and the BT reduced-order model are respectively used for testing in the model reduction benchmark example BIPS / 97 of the power system small-signal stability analysis.

[0190] For the preset time interval [0, T] = [0, 30], when the input vector u(t) = e -0.1t sin(t), the present application takes the Laguerre expansion coefficient l = 10, makes r = l = 10, and obtains a 10th-order LRBT reduced-order model. An existing balanced truncation method is used to obtain a BT reduced-order model with r = 10. In the case of the input vector u(t) = e -0.1t sin(t), the responses of the power system are predicted by the LRBT reduced-order model and the BT model respectively. The time-domain response vectors output by the two models are as Figure 3 shown, and the absolute errors between the time-domain response vectors of the two models and the actual response vector of the power system can be as Figure 4 shown.

[0191] For the preset time interval [0, T] = [0, 30], when the input vector u(t) = sin(0.5t), the present application takes the Laguerre expansion coefficient l = 20, makes r = l = 20, and obtains a 20th-order LRBT reduced-order model. An existing balanced truncation method is used to obtain a BT reduced-order model with r = 20. In the case of the input vector u(t) = sin(0.5t), the responses of the power system are predicted by the LRBT reduced-order model and the BT model respectively. The time-domain response vectors output by the two models are as Figure 5As shown, the absolute error between the time-domain response vectors of the two models and the actual response vector of the power system can be as Figure 6 shown.

[0192] In this test, when predicting the response of the power system using the method provided in this application and using the existing balanced truncation method to predict the response of the power system, the comparison of the time consumption can be as Figure 7 shown. From Figure 3-7 it can be seen that both this application and the existing balanced truncation method can well approximate the transient response of the power system. In addition, in the time domain, the method provided in this application has a better approximation effect than the classical balanced truncation method, and a reduced-order system with a smaller order and higher accuracy can be obtained. In terms of running time, compared with the classical balanced truncation method, the method provided in this application has improved the operation speed by about 98% and has higher computational efficiency. Thus, this application can balance a relatively high prediction accuracy and a relatively low computational complexity.

[0193] Next, the power system time-domain model reduction device based on Laguerre polynomials and Gram matrices provided in the embodiments of this application will be described. The power system time-domain model reduction device based on Laguerre polynomials and Gram matrices described below can be correspondingly referred to the power system time-domain model reduction method based on Laguerre polynomials and Gram matrices described above.

[0194] In one embodiment, as Figure 8 shown, this application provides a power system time-domain model reduction device 200 based on Laguerre polynomials and Gram matrices, including:

[0195] A state-space model acquisition module 202, configured to acquire the state-space model of the power system;

[0196] A Laguerre coefficient acquisition module 204, configured to use the Laguerre polynomial approximation algorithm to calculate each first Laguerre coefficient corresponding to the state impulse response of the state-space model, and each second Laguerre coefficient corresponding to the state impulse response of the dual model; wherein, the dual model is dual to the state-space model;

[0197] A low-rank decomposition module 206, configured to calculate a first low-rank decomposition factor matrix corresponding to the controllability Gram matrix according to each of the first Laguerre coefficients, and calculate a second low-rank decomposition factor matrix corresponding to the observability Gram matrix according to each of the second Laguerre coefficients;

[0198] A reduction module 208, configured to obtain a reduced-order model corresponding to the state space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix;

[0199] A response prediction module 210, configured to receive a to-be-predicted input vector of the power system in response to a system response prediction request, and predict a time-domain response vector of the power system according to the to-be-predicted input vector and the reduced-order model.

[0200] In one embodiment, the state space model includes a system matrix, an input matrix, and an output matrix. The Laguerre coefficient acquisition module 204 of the present application includes:

[0201] A first coefficient calculation unit, configured to calculate each of the first Laguerre coefficients based on the following expression:

[0202]

[0203] wherein, f0 is the coefficient of the first Laguerre expansion term corresponding to the state impulse response of the state space model; f i is the coefficient of the (i + 1)-th Laguerre expansion term corresponding to the state impulse response of the state space model; is an n1-dimensional identity matrix, n1 is the number of differential variables in the linear differential-algebraic equation of the power system, and the state space model is an equivalent model of the linear differential-algebraic equation; A is the system matrix, B is the input matrix, and l is the number of pre-determined Laguerre expansion terms;

[0204] A second coefficient calculation unit, configured to calculate each of the second Laguerre coefficients based on the following expression:

[0205]

[0206] wherein, g0 is the coefficient of the first Laguerre expansion term corresponding to the state impulse response of the dual model; g i is the coefficient of the (i + 1)-th Laguerre expansion term corresponding to the state impulse response of the dual model; A T is the transpose matrix of A, C T is the transpose matrix of C, and C is the output matrix.

[0207] In one embodiment, the low-rank decomposition module 206 of the present application includes:

[0208] An integral matrix construction unit, configured to calculate an integral result of the product of Laguerre polynomials within a preset time interval, and construct an integral matrix according to the integral result;

[0209] A Cholesky decomposition unit for performing Cholesky decomposition on the integral matrix to obtain a first lower triangular matrix;

[0210] A first low-rank decomposition unit for obtaining the first low-rank decomposition factor matrix according to the first lower triangular matrix and each of the first Laguerre coefficients;

[0211] A second low-rank decomposition unit for obtaining the second low-rank decomposition factor matrix according to the first lower triangular matrix and each of the second Laguerre coefficients.

[0212] In one embodiment, the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix are obtained based on the following expressions:

[0213]

[0214] In the formula, R is the first low-rank decomposition factor matrix; F is the coefficient matrix composed of each of the first Laguerre coefficients; J is the first lower triangular matrix; I p is the p-dimensional identity matrix, where p is the number of inputs of the power system; L is the second low-rank decomposition factor matrix; G is the coefficient matrix composed of each of the second Laguerre coefficients; I q is the q-dimensional identity matrix, where q is the number of outputs of the power system; is the Kronecker product operator.

[0215] In one embodiment, the order reduction module 208 of the present application includes:

[0216] An SVD decomposition unit for performing SVD decomposition on the L T R matrix, and respectively obtaining a left truncation matrix U r , a diagonal truncation matrix Σ r and a right truncation matrix V r according to the SVD decomposition result and a preset order reduction order; where R is the first low-rank decomposition factor matrix, and L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix;

[0217] A projection matrix calculation unit for respectively calculating a left projection matrix and a right projection matrix according to the left truncation matrix U r , the diagonal truncation matrix Σ r and the right truncation matrix V r ;

[0218] An order reduction unit for obtaining the order reduction model according to the left projection matrix and the right projection matrix.

[0219] In one embodiment, the projection matrix calculation unit of the present application is used to calculate the left projection matrix and the right projection matrix respectively based on the following expressions:

[0220]

[0221] In the formula, W l is the left projection matrix; Σ r is the diagonal truncation matrix; is the transpose matrix of U r , U r is the left truncation matrix; L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix; W r is the right projection matrix; R is the first low-rank decomposition factor matrix; V r is the right truncation matrix.

[0222] In one embodiment, the state space model includes a system matrix A, an input matrix B, an output matrix C, and a constant matrix D. The order reduction unit of the present application includes:

[0223] An order reduction matrix acquisition unit for respectively obtaining a reduced-order system matrix A r , a reduced-order input matrix B r and a reduced-order output matrix C r based on the following expressions:

[0224] A r =W l AW r , B r =W l B, C r =CW r , D r =D

[0225] In the formula, W l is the left projection matrix, and W r is the right projection matrix;

[0226] An order reduction model construction unit for constructing the order reduction model according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix, and the constant matrix.

[0227] In one embodiment, the present application further provides a storage medium storing computer-readable instructions, which when executed by one or more processors, cause the one or more processors to execute the steps of the power system time-domain model order reduction method based on Laguerre polynomials and Gram matrices in any embodiment.

[0228] In one embodiment, the present application further provides a computer device. Computer-readable instructions are stored in the computer device. When the computer-readable instructions are executed by one or more processors, the one or more processors are caused to execute the steps of the power system time-domain model reduction method based on Laguerre polynomials and Gram matrices as described in any of the embodiments.

[0229] Schematically, Figure 9 FIG. is an internal structural diagram of a computer device provided by an embodiment of the present application. In one example, the computer device may be a server. Referring to Figure 9 , computer device 900 includes a processing component 902, which further includes one or more processors, and memory resources represented by a memory 901 for storing instructions executable by the processing component 902, such as application programs. The application programs stored in the memory 901 may include one or more modules each corresponding to a set of instructions. In addition, the processing component 902 is configured to execute instructions to perform the steps of the power system time-domain model reduction method based on Laguerre polynomials and Gram matrices as described in any of the above embodiments.

[0230] Computer device 900 may further include a power supply component 903 configured to perform power management of computer device 900, a wired or wireless network interface 904 configured to connect computer device 900 to a network, and an input / output (I / O) interface 905. Computer device 900 may operate based on an operating system stored in memory 901, such as Windows Server TM, Mac OS XTM, Unix TM, Linux TM, Free BSDTM, or the like.

[0231] Those skilled in the art will appreciate that the internal structure of the computer device shown in the present application is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine certain components, or have a different component layout.

[0232] Finally, it should also be noted that in this text, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprise", "include" or any other variant thereof are intended to cover non-exclusive inclusion, such that a process, method, article or device comprising a series of elements not only includes those elements but also other elements not expressly listed, or elements inherent to such process, method, article or device. Without further limitation, an element defined by the statement "comprising an..." does not exclude the presence of additional identical elements in the process, method, article or device comprising said element. In this text, "a", "an", "the", "said" and "its" may also include the plural form, unless the context clearly indicates otherwise. "Plural" means at least two cases, such as 2, 3, 5 or 8, etc. "And / or" includes any and all combinations of the related listed items.

[0233] The various embodiments in this specification are described in a progressive manner. Each embodiment focuses on the differences from other embodiments. The various embodiments can be combined as needed, and the same or similar parts can be referred to each other.

[0234] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present application. Various modifications to these embodiments will be obvious to those skilled in the art. The general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to these embodiments shown herein, but rather will be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A time-domain model reduction method for power systems based on Laguerre polynomials and Gram matrices, characterized in that, Including: Obtain the state - space model of the power system; The state - space model includes a system matrix A, an input matrix B, an output matrix C, and a constant matrix D; Adopt the Laguerre polynomial approximation algorithm to calculate respectively each first Laguerre coefficient corresponding to the state impulse response of the state - space model, and each second Laguerre coefficient corresponding to the state impulse response of the dual model; wherein, the dual model is dual to the state - space model; Calculate the first low - rank decomposition factor matrix corresponding to the controllability Gram matrix according to each of the first Laguerre coefficients, and calculate the second low - rank decomposition factor matrix corresponding to the observability Gram matrix according to each of the second Laguerre coefficients; Based on the first low - rank decomposition factor matrix and the second low - rank decomposition factor matrix, obtain the reduced - order model corresponding to the state - space model; In response to a system response prediction request, receive the input vector to be predicted of the power system, and predict the time - domain response vector of the power system according to the input vector to be predicted and the reduced - order model; Wherein, the obtaining the reduced - order model corresponding to the state - space model based on the first low - rank decomposition factor matrix and the second low - rank decomposition factor matrix includes: Perform SVD decomposition on the L T R matrix, and respectively obtain the left truncated matrix U r , the diagonal truncated matrix Σ r and the right truncated matrix V r according to the SVD decomposition result and the preset order reduction order; where R is the first low-rank decomposition factor matrix, and L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix; According to the left truncation matrix U r , the diagonal truncation matrix Σ r and the right truncation matrix V r , the left projection matrix and the right projection matrix are respectively obtained based on the following expressions: where, W l is the left projection matrix; Σ r is the diagonal truncation matrix; is the transpose matrix of U r , U r is the left truncation matrix; L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix; W r is the right projection matrix; R is the first low-rank decomposition factor matrix; V r is the right truncation matrix; The reduced-order system matrix A, the reduced-order input matrix B, and the reduced-order output matrix C are obtained respectively based on the following expressions: r and r and r : A r = W l AW r , B r = W l B, C r = CW r , D r = D where W l is the left projection matrix, and W r is the right projection matrix; Construct the reduced - order model according to the reduced - order system matrix, the reduced - order input matrix, the reduced - order output matrix, and the constant matrix.

2. The method according to claim 1, wherein The state - space model includes a system matrix, an input matrix, and an output matrix; The adopting the Laguerre polynomial approximation algorithm to calculate respectively each first Laguerre coefficient corresponding to the state impulse response of the state - space model, and each second Laguerre coefficient corresponding to the state impulse response of the dual model includes: Based on the following expression, calculate each of the first Laguerre coefficients: where, f0 is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the state space model; f i is the coefficient of the (i + 1)-th Laguerre expansion corresponding to the state impulse response of the state space model; is an n1-dimensional identity matrix, n1 is the number of differential variables in the linear differential-algebraic equation of the power system, and the state space model is an equivalent model of the linear differential-algebraic equation; A is the system matrix, B is the input matrix, and l is the number of predetermined Laguerre expansion terms; Based on the following expression, calculate each of the second Laguerre coefficients: where \(g_0\) is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the dual model; \(g\) i is the coefficient of the \((i + 1)\)-th Laguerre expansion corresponding to the state impulse response of the dual model; \(A\) T is the transpose matrix of \(A\), \(C\) T is the transpose matrix of \(C\), and \(C\) is the output matrix.

3. The method according to claim 1, characterized in that The calculating the first low - rank decomposition factor matrix corresponding to the controllability Gram matrix according to each of the first Laguerre coefficients, and calculating the second low - rank decomposition factor matrix corresponding to the observability Gram matrix according to each of the second Laguerre coefficients includes: Calculate the integral result of the product of Laguerre polynomials within a preset time interval, and construct an integral matrix according to the integral result; Perform Cholesky decomposition on the integral matrix to obtain a first lower triangular matrix; According to the first lower triangular matrix and each of the first Laguerre coefficients, obtain the first low - rank decomposition factor matrix; According to the first lower triangular matrix and each of the second Laguerre coefficients, obtain the second low - rank decomposition factor matrix.

4. The method according to claim 3, wherein The first low - rank decomposition factor matrix and the second low - rank decomposition factor matrix are obtained based on the following expression: wherein, R is the first low-rank decomposition factor matrix; F is the coefficient matrix composed of each of the first Laguerre coefficients; J is the first lower triangular matrix; I p is a p-dimensional identity matrix, where p is the number of inputs of the power system; L is the second low-rank decomposition factor matrix; G is the coefficient matrix composed of each of the second Laguerre coefficients; I q is a q-dimensional identity matrix, where q is the number of outputs of the power system; is the Kronecker product operator.

5. A time-domain model reduction device for a power system based on Laguerre polynomials and Gram matrices, characterized in that, Including: A state - space model acquisition module for obtaining the state - space model of the power system; The state - space model includes a system matrix A, an input matrix B, an output matrix C, and a constant matrix D; The Laguerre coefficient acquisition module is used to calculate each first Laguerre coefficient corresponding to the state impulse response of the state space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model respectively by using the Laguerre polynomial approximation algorithm; wherein, the dual model is dual to the state space model; The low-rank decomposition module is used to calculate the first low-rank decomposition factor matrix corresponding to the controllable Gram matrix according to each of the first Laguerre coefficients and calculate the second low-rank decomposition factor matrix corresponding to the observable Gram matrix according to each of the second Laguerre coefficients; The order reduction module is used to obtain the reduced-order model corresponding to the state space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix; The response prediction module is used to, in response to a system response prediction request, receive the input vector to be predicted of the power system and predict the time-domain response vector of the power system according to the input vector to be predicted and the reduced-order model; Wherein, the order reduction module includes: The SVD decomposition unit is used to perform SVD decomposition on the L T R matrix, and respectively obtain the left truncated matrix U r , the diagonal truncated matrix Σ r and the right truncated matrix V r according to the SVD decomposition result and the preset reduced order; wherein, R is the first low-rank decomposition factor matrix, and L T is the transpose matrix of L, and L is the second low-rank decomposition factor matrix; A projection matrix calculation unit for obtaining a left projection matrix and a right projection matrix respectively based on the following expressions according to the left truncation matrix U r , the diagonal truncation matrix Σ r and the right truncation matrix V r : Where, W l is the left projection matrix; Σ r is the diagonal truncation matrix; is the transpose matrix of U r , and U r is the left truncation matrix; L T is the transpose matrix of L, and L is the second low-rank factorization factor matrix; W r is the right projection matrix; R is the first low-rank factorization factor matrix; V r is the right truncation matrix; A reduced-order matrix obtaining unit, configured to respectively obtain a reduced-order system matrix A r , a reduced-order input matrix B r and a reduced-order output matrix C r : A r = W l AW r , B r = W l B, C r = CW r , D r = D where W l is the left projection matrix, and W r is the right projection matrix; The reduced-order model construction unit is used to construct the reduced-order model according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix and the constant matrix.

6. A storage medium, characterized in that, The computer-readable instructions are stored in the storage medium, and when the computer-readable instructions are executed by one or more processors, the one or more processors are caused to execute the steps of the power system time-domain model order reduction method based on Laguerre polynomials and Gram matrices according to any one of claims 1 to 4.

7. A computer device, characterized in that, Including: One or more processors, and a memory; The computer-readable instructions are stored in the memory, and when the computer-readable instructions are executed by the one or more processors, the steps of the power system time-domain model order reduction method based on Laguerre polynomials and Gram matrices according to any one of claims 1 to 4 are executed.

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